2 + 2 = 4 a 2d/2d unitary/non-unitary correspondence
4 affiliations
- 1Stony Brook University
- 2School of Natural Sciences, Institute for Advanced Study
- 3University of Haifa at Oranim
- 4University of Haifa
Abstract
A bstract : Motivated by the observation that 2 + 2 = 4, we consider four-dimensional 𝒩 N = 2 superconformal field theories on S ² × Σ, turning on a suitable rigid supergravity background. On the one hand, reduction of a four-dimensional theory T on a Riemann surface Σ leads to a family ℱ[T,Σ ] F T Σ of two-dimensional (2, 2) unitary SCFTs, a two-dimensional analog of the four-dimensional theories of class 𝒮 S . On the other hand, reduction on S ² yields a non-unitary two-dimensional CFT 𝒞[T] C T whose chiral algebra is the same as the one associated to T by the standard SCFT/VOA correspondence. This construction upgrades the vertex operator algebra to a full-fledged two-dimensional CFT. What’s more, it leads to a novel 2d/2d correspondence, a “2 + 2 = 4” analog of the “4 + 2 = 6” AGT correspondence: the S ² partition function of ℱ[T;Σ ] F T Σ is computed by correlation functions of 𝒞[T] C T on Σ. The elliptic genus of ℱ[T;Σ ] F T Σ is instead computed by a topological QFT ℰ[T] E T on Σ. A central question is whether one can give a purely two-dimensional presentation of the family ℱ[T;Σ ] F T Σ of (2, 2) theories. We propose an algorithm to realize the (2, 2) theories as gauged linear sigma models when T is an Argyres-Douglas theory of type ( A ₁ , A 2 k ) and Σ an n -punctured sphere. We perform stringent checks of our conjecture for k = 1 and k = 2.
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BibTeX
@article{rastelli20262d2d,
title = {{2 + 2 = 4 a 2d/2d unitary/non-unitary correspondence}},
author = {Leonardo Rastelli and Brandon C. Rayhaun and Matteo Sacchi and Gabi Zafrir},
journal = {Journal of High Energy Physics},
year = {2026},
month = sep,
doi = {10.1007/jhep09(2026)242},
url = {https://link.springer.com/10.1007/JHEP09(2026)242},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - 2 + 2 = 4 a 2d/2d unitary/non-unitary correspondence
AU - Leonardo Rastelli
AU - Brandon C. Rayhaun
AU - Matteo Sacchi
AU - Gabi Zafrir
PY - 2026
DA - 2026/09/23
JO - Journal of High Energy Physics
DO - 10.1007/jhep09(2026)242
UR - https://link.springer.com/10.1007/JHEP09(2026)242
AB - A bstract : Motivated by the observation that 2 + 2 = 4, we consider four-dimensional 𝒩 N = 2 superconformal field theories on S ² × Σ, turning on a suitable rigid supergravity background. On the one hand, reduction of a four-dimensional theory T on a Riemann surface Σ leads to a family ℱ[T,Σ ] F T Σ of two-dimensional (2, 2) unitary SCFTs, a two-dimensional analog of the four-dimensional theories of class 𝒮 S . On the other hand, reduction on S ² yields a non-unitary two-dimensional CFT 𝒞[T] C T whose chiral algebra is the same as the one associated to T by the standard SCFT/VOA correspondence. This construction upgrades the vertex operator algebra to a full-fledged two-dimensional CFT. What’s more, it leads to a novel 2d/2d correspondence, a “2 + 2 = 4” analog of the “4 + 2 = 6” AGT correspondence: the S ² partition function of ℱ[T;Σ ] F T Σ is computed by correlation functions of 𝒞[T] C T on Σ. The elliptic genus of ℱ[T;Σ ] F T Σ is instead computed by a topological QFT ℰ[T] E T on Σ. A central question is whether one can give a purely two-dimensional presentation of the family ℱ[T;Σ ] F T Σ of (2, 2) theories. We propose an algorithm to realize the (2, 2) theories as gauged linear sigma models when T is an Argyres-Douglas theory of type ( A ₁ , A 2 k ) and Σ an n -punctured sphere. We perform stringent checks of our conjecture for k = 1 and k = 2.
SN - 1029-8479
ER -